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            <h1 style="display: none">CodeForces#752 Div3</h1>
            
              <p class="note note-info">
                
                  本文最后更新于：2 个月前
                
              </p>
            
            <div class="markdown-body" id="post-body">
              <p>传送门：<a target="_blank" rel="noopener" href="https://codeforces.com/contest/1619">https://codeforces.com/contest/1619</a></p>
<a id="more"></a>
<h2 id="A-Square-String"><a href="#A-Square-String" class="headerlink" title="A. Square String?"></a>A. Square String?</h2><p>对半匹配</p>
<div class='spoiler collapsed'>
    <div class='spoiler-title'>
        Code
    </div>
    <div class='spoiler-content'>
        <div class="hljs"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;map&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;set&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cmath&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;queue&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;bitset&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;vector&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstdio&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstring&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstdlib&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;iostream&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;algorithm&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> rep(i,a,b) for(int i = (a);i &lt;= (b);++i)</span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> per(i,a,b) for(int i = (a);i &gt;= (b);--i) </span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> mkp std::make_pair</span>
<span class="hljs-keyword">typedef</span> <span class="hljs-keyword">long</span> <span class="hljs-keyword">long</span> ll;
<span class="hljs-keyword">typedef</span> <span class="hljs-keyword">unsigned</span> <span class="hljs-keyword">long</span> <span class="hljs-keyword">long</span> ull;
<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">string</span>;<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">cin</span>;<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">cout</span>;
<span class="hljs-function"><span class="hljs-keyword">inline</span> <span class="hljs-keyword">bool</span> <span class="hljs-title">cmp</span><span class="hljs-params">(<span class="hljs-keyword">int</span> x,<span class="hljs-keyword">int</span> y)</span></span>&#123;<span class="hljs-keyword">return</span> x &lt; y;&#125;
<span class="hljs-comment">/* -fsanitize=undefined  */</span>
<span class="hljs-keyword">const</span> <span class="hljs-keyword">int</span> N = <span class="hljs-number">1e6</span>+<span class="hljs-number">9</span>;
<span class="hljs-keyword">const</span> <span class="hljs-keyword">int</span> inf = <span class="hljs-number">1e9</span>+<span class="hljs-number">9</span>;
<span class="hljs-keyword">const</span> <span class="hljs-keyword">double</span> eps = <span class="hljs-number">1e-7</span>;
<span class="hljs-keyword">int</span> _,n,a[<span class="hljs-number">2</span>*N];
<span class="hljs-built_in">string</span> str;

<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span>&#123;
    <span class="hljs-built_in">std</span>::ios::sync_with_stdio(<span class="hljs-number">0</span>);<span class="hljs-built_in">cin</span>.tie(<span class="hljs-number">0</span>);<span class="hljs-built_in">cout</span>.tie(<span class="hljs-number">0</span>);
    <span class="hljs-meta">#<span class="hljs-meta-keyword">ifdef</span> LOCAL <span class="hljs-comment">//  -DLOCAL</span></span>
        freopen(<span class="hljs-string">&quot;in.in&quot;</span>, <span class="hljs-string">&quot;r&quot;</span>, <span class="hljs-built_in">stdin</span>);
    <span class="hljs-meta">#<span class="hljs-meta-keyword">endif</span></span>
    <span class="hljs-built_in">cin</span> &gt;&gt; _;
    <span class="hljs-keyword">while</span>(_--)&#123;
        <span class="hljs-built_in">cin</span> &gt;&gt; str;
        n = str.length();
        <span class="hljs-keyword">if</span>(n%<span class="hljs-number">2</span>)&#123;
            <span class="hljs-built_in">cout</span> &lt;&lt; <span class="hljs-string">&quot;NO\n&quot;</span>;
            <span class="hljs-keyword">continue</span>;
        &#125; <span class="hljs-keyword">else</span> &#123;
            <span class="hljs-keyword">bool</span> flag = <span class="hljs-literal">true</span>;
            rep(i,<span class="hljs-number">1</span>,n/<span class="hljs-number">2</span>)&#123;
                <span class="hljs-keyword">if</span>(str[i<span class="hljs-number">-1</span>] == str[i<span class="hljs-number">-1</span>+n/<span class="hljs-number">2</span>]) <span class="hljs-keyword">continue</span>;
                flag = <span class="hljs-literal">false</span>;
                <span class="hljs-keyword">break</span>;
            &#125;
            <span class="hljs-keyword">if</span>(flag) <span class="hljs-built_in">cout</span> &lt;&lt; <span class="hljs-string">&quot;YES\n&quot;</span>;
            <span class="hljs-keyword">else</span> <span class="hljs-built_in">cout</span> &lt;&lt; <span class="hljs-string">&quot;NO\n&quot;</span>;
        &#125;
    &#125;
    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;
&#125;
</code></pre></div>

    </div>
</div>
<hr>
<h2 id="B-Squares-and-Cubes"><a href="#B-Squares-and-Cubes" class="headerlink" title="B. Squares and Cubes"></a>B. Squares and Cubes</h2><p>找<code>1~n</code>中<strong>完全平方数</strong>或者<strong>完全立方数</strong>的数量，先预处理所有完全平方数数组和完全立方数数组。</p>
<p>然后遍历小于<code>n</code>的完全立方数数量<code>cnt1</code>，找到其中<strong>也是</strong>完全平方数的数量<code>cnt2</code>，再找到所有小于<code>n</code>的完全平方数数量<code>cnt3</code>，显然，答案就是<code>cnt1</code> + <code>cnt3</code> - <code>cnt2</code>。</p>
<div class='spoiler collapsed'>
    <div class='spoiler-title'>
        Code
    </div>
    <div class='spoiler-content'>
        <div class="hljs"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;map&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;set&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cmath&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;queue&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;bitset&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;vector&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstdio&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstring&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstdlib&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;iostream&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;algorithm&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> rep(i,a,b) for(int i = (a);i &lt;= (b);++i)</span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> per(i,a,b) for(int i = (a);i &gt;= (b);--i) </span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> mkp std::make_pair</span>
<span class="hljs-keyword">typedef</span> <span class="hljs-keyword">long</span> <span class="hljs-keyword">long</span> ll;
<span class="hljs-keyword">typedef</span> <span class="hljs-keyword">unsigned</span> <span class="hljs-keyword">long</span> <span class="hljs-keyword">long</span> ull;
<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">string</span>;<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">cin</span>;<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">cout</span>;
<span class="hljs-function"><span class="hljs-keyword">inline</span> <span class="hljs-keyword">bool</span> <span class="hljs-title">cmp</span><span class="hljs-params">(<span class="hljs-keyword">int</span> x,<span class="hljs-keyword">int</span> y)</span></span>&#123;<span class="hljs-keyword">return</span> x &lt; y;&#125;
<span class="hljs-comment">/* -fsanitize=undefined  */</span>
<span class="hljs-keyword">const</span> <span class="hljs-keyword">int</span> N = <span class="hljs-number">1e6</span>+<span class="hljs-number">9</span>;
<span class="hljs-keyword">const</span> <span class="hljs-keyword">int</span> inf = <span class="hljs-number">1e9</span>+<span class="hljs-number">9</span>;
<span class="hljs-keyword">const</span> <span class="hljs-keyword">double</span> eps = <span class="hljs-number">1e-7</span>;
<span class="hljs-keyword">int</span> _,n,a2[<span class="hljs-number">2</span>*N],l2,a3[<span class="hljs-number">2</span>*N],l3;
<span class="hljs-built_in">std</span>::<span class="hljs-built_in">set</span>&lt;<span class="hljs-keyword">int</span>&gt; s3,s2;

<span class="hljs-function"><span class="hljs-keyword">void</span> <span class="hljs-title">init</span><span class="hljs-params">()</span></span>&#123;
    rep(i,<span class="hljs-number">1</span>,inf)&#123;
        <span class="hljs-keyword">if</span>(i * i &gt;= inf) <span class="hljs-keyword">break</span>;
        a2[++l2] = i*i;
        s2.insert(i*i);
    &#125;
    rep(i,<span class="hljs-number">1</span>,inf)&#123;
        <span class="hljs-keyword">if</span>(i * i * i &gt;= inf) <span class="hljs-keyword">break</span>;
        a3[++l3] = i*i*i;
        <span class="hljs-comment">// s3.insert(i*i*i);</span>
    &#125;
&#125;

<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span>&#123;
    <span class="hljs-built_in">std</span>::ios::sync_with_stdio(<span class="hljs-number">0</span>);<span class="hljs-built_in">cin</span>.tie(<span class="hljs-number">0</span>);<span class="hljs-built_in">cout</span>.tie(<span class="hljs-number">0</span>);
    <span class="hljs-meta">#<span class="hljs-meta-keyword">ifdef</span> LOCAL <span class="hljs-comment">//  -DLOCAL</span></span>
        freopen(<span class="hljs-string">&quot;in.in&quot;</span>, <span class="hljs-string">&quot;r&quot;</span>, <span class="hljs-built_in">stdin</span>);
    <span class="hljs-meta">#<span class="hljs-meta-keyword">endif</span></span>
    <span class="hljs-built_in">cin</span> &gt;&gt; _;
    init();
    <span class="hljs-keyword">while</span>(_--)&#123;
        <span class="hljs-built_in">cin</span> &gt;&gt; n;
        <span class="hljs-keyword">int</span> cnt = <span class="hljs-number">0</span>;
        rep(i,<span class="hljs-number">1</span>,l3)&#123;
            <span class="hljs-keyword">if</span>(a3[i] &lt;= n) ++cnt;
            <span class="hljs-comment">// if(a3[i] &lt;= n) cout &lt;&lt; a3[i] &lt;&lt; &quot;\n&quot;;</span>
            <span class="hljs-keyword">if</span>(a3[i] &lt;= n &amp;&amp; s2.find( a3[i] ) != s2.end()) --cnt;
        &#125;
        rep(i,<span class="hljs-number">1</span>,l2)&#123;
            <span class="hljs-keyword">if</span>(a2[i] &lt;= n) ++cnt;
            <span class="hljs-comment">// if(a2[i] &lt;= n) cout &lt;&lt; a2[i] &lt;&lt; &quot;\n&quot;;</span>
        &#125;
        <span class="hljs-built_in">cout</span> &lt;&lt; cnt &lt;&lt; <span class="hljs-string">&quot;\n&quot;</span>;
    &#125;
    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;
&#125;
</code></pre></div>

    </div>
</div>
<hr>
<h2 id="C-Wrong-Addition"><a href="#C-Wrong-Addition" class="headerlink" title="C. Wrong Addition"></a>C. Wrong Addition</h2><p>实际上就是把进位分离出来的一个竖式加法，我们要去做这种模式下的一个减法。那做法就很简单了，减出负数就进位就可以了。</p>
<div class='spoiler collapsed'>
    <div class='spoiler-title'>
        Code
    </div>
    <div class='spoiler-content'>
        <div class="hljs"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;map&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;set&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cmath&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;queue&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;bitset&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;vector&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstdio&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstring&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstdlib&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;iostream&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;algorithm&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> rep(i,a,b) for(int i = (a);i &lt;= (b);++i)</span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> per(i,a,b) for(int i = (a);i &gt;= (b);--i) </span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> mkp std::make_pair</span>
<span class="hljs-keyword">typedef</span> <span class="hljs-keyword">long</span> <span class="hljs-keyword">long</span> ll;
<span class="hljs-keyword">typedef</span> <span class="hljs-keyword">unsigned</span> <span class="hljs-keyword">long</span> <span class="hljs-keyword">long</span> ull;
<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">string</span>;<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">cin</span>;<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">cout</span>;
<span class="hljs-function"><span class="hljs-keyword">inline</span> <span class="hljs-keyword">bool</span> <span class="hljs-title">cmp</span><span class="hljs-params">(<span class="hljs-keyword">int</span> x,<span class="hljs-keyword">int</span> y)</span></span>&#123;<span class="hljs-keyword">return</span> x &lt; y;&#125;
<span class="hljs-comment">/* -fsanitize=undefined  */</span>
<span class="hljs-keyword">const</span> <span class="hljs-keyword">int</span> N = <span class="hljs-number">1e6</span>+<span class="hljs-number">9</span>;
<span class="hljs-keyword">const</span> <span class="hljs-keyword">int</span> inf = <span class="hljs-number">1e9</span>+<span class="hljs-number">9</span>;
<span class="hljs-keyword">const</span> <span class="hljs-keyword">double</span> eps = <span class="hljs-number">1e-7</span>;
<span class="hljs-keyword">int</span> _,n,lenA,lenS,b[<span class="hljs-number">2</span>*N],a[<span class="hljs-number">2</span>*N],s[<span class="hljs-number">2</span>*N];
<span class="hljs-built_in">string</span> A,S;

<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span>&#123;
    <span class="hljs-built_in">std</span>::ios::sync_with_stdio(<span class="hljs-number">0</span>);<span class="hljs-built_in">cin</span>.tie(<span class="hljs-number">0</span>);<span class="hljs-built_in">cout</span>.tie(<span class="hljs-number">0</span>);
    <span class="hljs-meta">#<span class="hljs-meta-keyword">ifdef</span> LOCAL <span class="hljs-comment">//  -DLOCAL</span></span>
        freopen(<span class="hljs-string">&quot;in.in&quot;</span>, <span class="hljs-string">&quot;r&quot;</span>, <span class="hljs-built_in">stdin</span>);
    <span class="hljs-meta">#<span class="hljs-meta-keyword">endif</span></span>
    <span class="hljs-built_in">cin</span> &gt;&gt; _;
    <span class="hljs-keyword">while</span>(_--)&#123;
        <span class="hljs-built_in">cin</span> &gt;&gt; A &gt;&gt; S;
        lenA = A.length() , lenS = S.length();
        rep(i,<span class="hljs-number">0</span>,lenA<span class="hljs-number">-1</span>) a[lenA-i] = A[i] - <span class="hljs-string">&#x27;0&#x27;</span>;
        rep(i,<span class="hljs-number">0</span>,lenS<span class="hljs-number">-1</span>) s[lenS-i] = S[i] - <span class="hljs-string">&#x27;0&#x27;</span>;
        rep(i,<span class="hljs-number">0</span>,lenS+<span class="hljs-number">1</span>) b[i] = <span class="hljs-number">0</span>; 
        <span class="hljs-keyword">int</span> j = <span class="hljs-number">1</span> , i = <span class="hljs-number">1</span> , p;
        <span class="hljs-keyword">bool</span> ans = <span class="hljs-literal">true</span>;
        <span class="hljs-keyword">while</span>(<span class="hljs-number">1</span>)&#123;
            p = i;
            <span class="hljs-keyword">if</span>(j &gt; lenS) <span class="hljs-keyword">break</span>;
            <span class="hljs-keyword">if</span>(i &gt; lenA) a[i] = <span class="hljs-number">0</span>;
            <span class="hljs-keyword">if</span>(s[j] &gt;= a[i])&#123;
                b[i] = s[j] - a[i];
                j += <span class="hljs-number">1</span>;
            &#125; <span class="hljs-keyword">else</span>&#123;
                <span class="hljs-keyword">if</span>(s[j+<span class="hljs-number">1</span>] &gt; <span class="hljs-number">1</span> || s[j+<span class="hljs-number">1</span>] &lt; <span class="hljs-number">1</span>)&#123;
                    ans = <span class="hljs-literal">false</span>;
                    <span class="hljs-keyword">break</span>;
                &#125;
                b[i] = s[j] + s[j+<span class="hljs-number">1</span>] * <span class="hljs-number">10</span> - a[i];
                j += <span class="hljs-number">2</span>;
            &#125;
            ++i;
        &#125;
        <span class="hljs-keyword">if</span>(!ans || i &lt;= lenA)&#123;
            <span class="hljs-built_in">cout</span> &lt;&lt; <span class="hljs-string">&quot;-1\n&quot;</span>;
            <span class="hljs-keyword">continue</span>;
        &#125;
        <span class="hljs-keyword">bool</span> leadZero = <span class="hljs-literal">true</span>;
        <span class="hljs-keyword">bool</span> isZero = <span class="hljs-literal">true</span>;
        per(i,p,<span class="hljs-number">1</span>)&#123;
            <span class="hljs-keyword">if</span>(b[i]) leadZero = <span class="hljs-literal">false</span>;
            <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span>(leadZero) <span class="hljs-keyword">continue</span>;
            <span class="hljs-built_in">cout</span> &lt;&lt; b[i];
            isZero = <span class="hljs-literal">false</span>;
        &#125;
        <span class="hljs-keyword">if</span>(isZero) <span class="hljs-built_in">cout</span> &lt;&lt; <span class="hljs-string">&quot;-1&quot;</span>;
        <span class="hljs-built_in">cout</span> &lt;&lt; <span class="hljs-string">&quot;\n&quot;</span>;
    &#125;
    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;
&#125;</code></pre></div>

    </div>
</div>
<hr>
<h2 id="D-New-Year’s-Problem"><a href="#D-New-Year’s-Problem" class="headerlink" title="D. New Year’s Problem"></a>D. New Year’s Problem</h2><p>You can get English version <a target="_blank" rel="noopener" href="https://codeforces.com/blog/entry/98159?#comment-871441">here</a>.</p>
<p>官方给出的解法是二分答案，但是我在评论区看到一个更有意思的解法，公式如下：</p>
<script type="math/tex; mode=display">
ans = min\{ minTop1ByFriends , maxTop2ByShops  \}</script><p>解释如下：</p>
<p>根据题意，首先容易得到 $ans \leq minTop1ByFriends$​ </p>
<p>规定<code>a[i][j]</code>表示礼物<code>j</code>对人<code>i</code>的贡献</p>
<ol>
<li>让我们先假设$n-1 \leq m$​</li>
</ol>
<ul>
<li><p>下面给出一个买礼物的策略：</p>
<ul>
<li>选择某一个人<code>I</code>，找到<strong>他最喜欢</strong>的礼物<code>J</code>，并且先<u>假设<code>a[I][J]</code> = <code>max&#123; a[1][J] , a[2][J] , ... , a[n][J] &#125;</code>（即这个礼物最被<code>I</code>喜欢）</u>，我们称之为<strong>假设A</strong></li>
<li><p>找到<code>a[1][J] , a[2][J] , ... , a[n][J]</code>中的<strong>次</strong>大项（Top2），并找到该对应的<code>I&#39;</code>，于是已经有2个人的礼物被安排好了</p>
<ul>
<li>这两个人的<code>joy</code>分别是<code>top1ByFriends[I]</code>和<code>top2ByShops[J]</code></li>
</ul>
</li>
<li><p>对于剩下的<code>n-2</code>个人，必定能找到他们最喜欢的礼物</p>
<ul>
<li>对于这些人，他们的<code>joy</code>分别是<code>top1ByFriends[i]</code></li>
</ul>
</li>
<li>不难得到，在这种策略下，遍历所有的“某一个人<code>I</code>”，可以证明，对于某一个确定的<code>I</code>，在<strong>假设A</strong>成立的情况下，这种策略是<strong>可行</strong>的，而在这些所有策略里找最大值，容易得到$ans\leq maxTop2ByShops$​​。​</li>
<li>现在我们来考虑<strong>假设A</strong>不成立的情况，显然，这种情况不影响我们的结论，因为此时<code>top1ByFriends[I]</code>&lt;<code>top2ByShops[J]</code>，答案就主要由<code>top1ByFriends[I]</code>决定，我们不再需要关心<code>I&#39;</code>到底是否是次大还是最大，只要比<code>I</code>的大就行了</li>
</ul>
</li>
<li><p>所以</p>
<script type="math/tex; mode=display">
ans \leq min\{minTop1ByFriends , maxTop2ByShops  \}</script></li>
<li><p>我们再来看取等问题</p>
<ul>
<li>当$minTop1ByFriends \leq maxTop2ByShops$时，我们取<code>I</code>为这个取到<code>minTop1ByFriens</code>的<code>I</code>，按照上述策略，可取等</li>
<li>当$minTop1ByFriends \geq maxTop2ByShops$时，我们取<code>J</code>为这个取到<code>maxTop2ByShops</code>的<code>J</code>，并选取该<code>J</code>对应的<code>Top1ByShops</code>对应的<code>I</code>，按照上述策略，可取等</li>
</ul>
</li>
<li><p>所以可以取等，所以</p>
<script type="math/tex; mode=display">
ans = min\{ minTop1ByFriends , maxTop2ByShops  \}</script></li>
</ul>
<ol>
<li>对于$n-1&gt; m$的情况，我在<strong><a target="_blank" rel="noopener" href="https://codeforces.com/blog/entry/98159?#comment-871443">这里</a></strong>做了详细解释</li>
</ol>
<div class='spoiler collapsed'>
    <div class='spoiler-title'>
        Code
    </div>
    <div class='spoiler-content'>
        <div class="hljs"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;map&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;set&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cmath&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;queue&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;bitset&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;vector&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstdio&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstring&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstdlib&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;iostream&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;algorithm&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> rep(i,a,b) for(int i = (a);i &lt;= (b);++i)</span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> per(i,a,b) for(int i = (a);i &gt;= (b);--i) </span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> mkp std::make_pair</span>
<span class="hljs-keyword">typedef</span> <span class="hljs-keyword">long</span> <span class="hljs-keyword">long</span> ll;
<span class="hljs-keyword">typedef</span> <span class="hljs-keyword">unsigned</span> <span class="hljs-keyword">long</span> <span class="hljs-keyword">long</span> ull;
<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">string</span>;<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">cin</span>;<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">cout</span>;
<span class="hljs-function"><span class="hljs-keyword">inline</span> <span class="hljs-keyword">bool</span> <span class="hljs-title">cmp</span><span class="hljs-params">(<span class="hljs-keyword">int</span> x,<span class="hljs-keyword">int</span> y)</span></span>&#123;<span class="hljs-keyword">return</span> x &lt; y;&#125;
<span class="hljs-comment">/* -fsanitize=undefined  */</span>
<span class="hljs-keyword">const</span> <span class="hljs-keyword">int</span> N = <span class="hljs-number">1e6</span>+<span class="hljs-number">9</span>;
<span class="hljs-keyword">const</span> <span class="hljs-keyword">int</span> inf = <span class="hljs-number">1e9</span>+<span class="hljs-number">9</span>;
<span class="hljs-keyword">const</span> <span class="hljs-keyword">double</span> eps = <span class="hljs-number">1e-7</span>;
<span class="hljs-keyword">int</span> _,n,m,a[<span class="hljs-number">2</span>*N],minTop1ByFriends,maxTop2ByShops,top1ByFriends[<span class="hljs-number">2</span>*N],top2ByShops[<span class="hljs-number">2</span>*N];

<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span>&#123;
    <span class="hljs-built_in">std</span>::ios::sync_with_stdio(<span class="hljs-number">0</span>);<span class="hljs-built_in">cin</span>.tie(<span class="hljs-number">0</span>);<span class="hljs-built_in">cout</span>.tie(<span class="hljs-number">0</span>);
    <span class="hljs-meta">#<span class="hljs-meta-keyword">ifdef</span> LOCAL <span class="hljs-comment">//  -DLOCAL</span></span>
        freopen(<span class="hljs-string">&quot;in.in&quot;</span>, <span class="hljs-string">&quot;r&quot;</span>, <span class="hljs-built_in">stdin</span>);
    <span class="hljs-meta">#<span class="hljs-meta-keyword">endif</span></span>
    <span class="hljs-built_in">cin</span> &gt;&gt; _;
    <span class="hljs-keyword">while</span>(_--)&#123;
        <span class="hljs-built_in">cin</span> &gt;&gt; m &gt;&gt; n;
        rep(j,<span class="hljs-number">1</span>,m) rep(i,<span class="hljs-number">1</span>,n) <span class="hljs-built_in">cin</span> &gt;&gt; a[(i<span class="hljs-number">-1</span>)*m + (j<span class="hljs-number">-1</span>)];
        minTop1ByFriends = inf , maxTop2ByShops = <span class="hljs-number">0</span>;
        rep(i,<span class="hljs-number">1</span>,n)&#123;
            top1ByFriends[i] = <span class="hljs-number">0</span>;
            rep(j,<span class="hljs-number">1</span>,m)&#123;
                <span class="hljs-keyword">if</span>(a[ (i<span class="hljs-number">-1</span>)*m + (j<span class="hljs-number">-1</span>) ] &gt; top1ByFriends[i]) top1ByFriends[i] = a[ (i<span class="hljs-number">-1</span>)*m + (j<span class="hljs-number">-1</span>) ];
            &#125;
        &#125;
        rep(j,<span class="hljs-number">1</span>,m)&#123;
            <span class="hljs-keyword">int</span> tmpMax = j<span class="hljs-number">-1</span>;
            rep(i,<span class="hljs-number">1</span>,n)&#123;
                <span class="hljs-keyword">if</span>(a[ (i<span class="hljs-number">-1</span>)*m + (j<span class="hljs-number">-1</span>) ] &gt; a[tmpMax]) tmpMax = (i<span class="hljs-number">-1</span>)*m + (j<span class="hljs-number">-1</span>);
            &#125;
            top2ByShops[j] = <span class="hljs-number">0</span>;
            rep(i,<span class="hljs-number">1</span>,n)&#123;
                <span class="hljs-keyword">if</span>((i<span class="hljs-number">-1</span>)*m + (j<span class="hljs-number">-1</span>) == tmpMax) <span class="hljs-keyword">continue</span>;
                <span class="hljs-keyword">if</span>(a[ (i<span class="hljs-number">-1</span>)*m + (j<span class="hljs-number">-1</span>) ] &gt; top2ByShops[j]) top2ByShops[j] = a[ (i<span class="hljs-number">-1</span>)*m + (j<span class="hljs-number">-1</span>) ];
            &#125;
        &#125;
        minTop1ByFriends = inf;
        rep(i,<span class="hljs-number">1</span>,n)&#123;
            minTop1ByFriends = <span class="hljs-built_in">std</span>::min(minTop1ByFriends,top1ByFriends[i]);
        &#125;
        maxTop2ByShops = <span class="hljs-number">0</span>;
        rep(j,<span class="hljs-number">1</span>,m)&#123;
            maxTop2ByShops = <span class="hljs-built_in">std</span>::max(maxTop2ByShops,top2ByShops[j]);
        &#125;
        <span class="hljs-comment">// cout &lt;&lt; &quot;debug::  &quot; &lt;&lt; minTop1ByFriends &lt;&lt; &quot;   &quot; &lt;&lt; maxTop2ByShops &lt;&lt; &quot;  \n&quot;;</span>
        <span class="hljs-built_in">cout</span> &lt;&lt; <span class="hljs-built_in">std</span>::min(minTop1ByFriends,maxTop2ByShops) &lt;&lt; <span class="hljs-string">&quot;\n&quot;</span>;
    &#125;
    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;
&#125;
</code></pre></div>

    </div>
</div>
<hr>
<h2 id="E-MEX-and-Increments"><a href="#E-MEX-and-Increments" class="headerlink" title="E. MEX and Increments"></a>E. MEX and Increments</h2><h4 id="1-首先解决有没有策略的问题"><a href="#1-首先解决有没有策略的问题" class="headerlink" title="1.首先解决有没有策略的问题"></a>1.首先解决有没有策略的问题</h4><p>只要存在至少<code>k</code>个小于<code>k</code>的数，就可以实现<code>MEX = k</code>，对于等于<code>k</code>的这些要通过把他们变大来让他们都不等于<code>k</code></p>
<h4 id="2-解决最优策略的问题"><a href="#2-解决最优策略的问题" class="headerlink" title="2.解决最优策略的问题"></a>2.解决最优策略的问题</h4><p>很容易理解可以通过递推实现，假设实现<code>MEX = k-1</code>最少需要<code>p</code>步，其中由<code>q</code>步用来将等于<code>k-1</code>的数变成<code>k</code>（即原数列中存在<code>q</code>个等于<code>k-1</code>的数），<code>q &gt;= 0</code>，等于<code>k</code>的数一共由<code>q&#39;</code>个（即需要<code>q&#39;</code>步来让它们变成<code>k+1</code>），那么实现<code>MEX = k</code>所需要的最少的步骤数即为<code>p - q + q&#39; + delP</code>，其中：</p>
<ul>
<li>如果<code>q = 0</code>，<code>delP</code>为<strong>最大</strong>的空闲的数</li>
<li>如果<code>q &gt; 0</code>，<code>delP = 0</code></li>
</ul>
<p>而所谓的空闲的数，就是在实现<code>MEX = k-1</code>后还剩下的多余的小于<code>k-1</code>的数</p>
<p>注意优化查找空闲的数这个步骤</p>
<div class='spoiler collapsed'>
    <div class='spoiler-title'>
        Code
    </div>
    <div class='spoiler-content'>
        <div class="hljs"><pre><code class="hljs cpp"><span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;map&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;set&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cmath&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;queue&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;bitset&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;vector&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstdio&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstring&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;cstdlib&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;iostream&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">include</span><span class="hljs-meta-string">&lt;algorithm&gt;</span></span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> rep(i,a,b) for(int i = (a);i &lt;= (b);++i)</span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> per(i,a,b) for(int i = (a);i &gt;= (b);--i) </span>
<span class="hljs-meta">#<span class="hljs-meta-keyword">define</span> mkp std::make_pair</span>
<span class="hljs-keyword">typedef</span> <span class="hljs-keyword">long</span> <span class="hljs-keyword">long</span> ll;
<span class="hljs-keyword">typedef</span> <span class="hljs-keyword">unsigned</span> <span class="hljs-keyword">long</span> <span class="hljs-keyword">long</span> ull;
<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">string</span>;<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">cin</span>;<span class="hljs-keyword">using</span> <span class="hljs-built_in">std</span>::<span class="hljs-built_in">cout</span>;
<span class="hljs-function"><span class="hljs-keyword">inline</span> <span class="hljs-keyword">bool</span> <span class="hljs-title">cmp</span><span class="hljs-params">(<span class="hljs-keyword">int</span> x,<span class="hljs-keyword">int</span> y)</span></span>&#123;<span class="hljs-keyword">return</span> x &lt; y;&#125;
<span class="hljs-comment">/* -fsanitize=undefined  */</span>
<span class="hljs-keyword">const</span> <span class="hljs-keyword">int</span> N = <span class="hljs-number">1e6</span>+<span class="hljs-number">9</span>;
<span class="hljs-keyword">const</span> <span class="hljs-keyword">int</span> inf = <span class="hljs-number">1e9</span>+<span class="hljs-number">9</span>;
<span class="hljs-keyword">const</span> <span class="hljs-keyword">double</span> eps = <span class="hljs-number">1e-7</span>;
<span class="hljs-keyword">int</span> _,n,a[<span class="hljs-number">2</span>*N],less[<span class="hljs-number">2</span>*N],equal[<span class="hljs-number">2</span>*N],<span class="hljs-built_in">stack</span>[<span class="hljs-number">2</span>*N],top;
ll ans;


<span class="hljs-function"><span class="hljs-keyword">int</span> <span class="hljs-title">main</span><span class="hljs-params">()</span></span>&#123;
    <span class="hljs-built_in">std</span>::ios::sync_with_stdio(<span class="hljs-number">0</span>);<span class="hljs-built_in">cin</span>.tie(<span class="hljs-number">0</span>);<span class="hljs-built_in">cout</span>.tie(<span class="hljs-number">0</span>);
    <span class="hljs-meta">#<span class="hljs-meta-keyword">ifdef</span> LOCAL <span class="hljs-comment">//  -DLOCAL</span></span>
        freopen(<span class="hljs-string">&quot;in.in&quot;</span>, <span class="hljs-string">&quot;r&quot;</span>, <span class="hljs-built_in">stdin</span>);
    <span class="hljs-meta">#<span class="hljs-meta-keyword">endif</span></span>
    <span class="hljs-built_in">cin</span> &gt;&gt; _;
    <span class="hljs-keyword">while</span>(_--)&#123;
        <span class="hljs-built_in">cin</span> &gt;&gt; n;
        rep(i,<span class="hljs-number">1</span>,n) <span class="hljs-built_in">cin</span> &gt;&gt; a[i];
        <span class="hljs-built_in">std</span>::sort(a+<span class="hljs-number">1</span>,a+n+<span class="hljs-number">1</span>,cmp);
        rep(i,<span class="hljs-number">0</span>,n+<span class="hljs-number">10</span>) less[i] = equal[i] = <span class="hljs-built_in">stack</span>[i] = <span class="hljs-number">0</span>;
        top = <span class="hljs-number">5</span>;
        <span class="hljs-keyword">int</span> p = <span class="hljs-number">1</span>;
        rep(i,<span class="hljs-number">0</span>,n)&#123;
            <span class="hljs-keyword">if</span>(i &gt; <span class="hljs-number">0</span>) less[i] += equal[i<span class="hljs-number">-1</span>] + less[i<span class="hljs-number">-1</span>];
            <span class="hljs-keyword">while</span>(p &lt;= n &amp;&amp; a[p] &lt; i) ++less[i],++p;
            <span class="hljs-keyword">while</span>(p &lt;= n &amp;&amp; a[p] == i) ++equal[i],++p;
            <span class="hljs-comment">// cout &lt;&lt; less[i] &lt;&lt; &quot; ; &quot;;</span>
        &#125;
        <span class="hljs-comment">// cout &lt;&lt; &quot;\n&quot;;</span>
        <span class="hljs-keyword">bool</span> unable = <span class="hljs-literal">false</span>;
        ans = <span class="hljs-number">0</span>;
        rep(i,<span class="hljs-number">0</span>,n)&#123;
            <span class="hljs-keyword">if</span>(unable)&#123;
                <span class="hljs-built_in">cout</span> &lt;&lt; <span class="hljs-string">&quot;-1 &quot;</span>;
            &#125; <span class="hljs-keyword">else</span> <span class="hljs-keyword">if</span>(less[i] &lt; i)&#123;
                unable = <span class="hljs-literal">true</span>;
                <span class="hljs-built_in">cout</span> &lt;&lt; <span class="hljs-string">&quot;-1 &quot;</span>;
            &#125; <span class="hljs-keyword">else</span> &#123;
                <span class="hljs-keyword">if</span>(i == <span class="hljs-number">0</span>) ans = equal[<span class="hljs-number">0</span>];
                <span class="hljs-keyword">else</span> &#123;
                    <span class="hljs-comment">// cout &lt;&lt; &quot;debug :: &quot; &lt;&lt; ans &lt;&lt; &quot; &quot; &lt;&lt; equal[i-1] &lt;&lt; &quot; &quot; &lt;&lt; equal[i] &lt;&lt; &quot; &quot;;</span>
                    <span class="hljs-keyword">if</span>(i &gt; <span class="hljs-number">1</span>)&#123;
                        <span class="hljs-keyword">if</span>(equal[i<span class="hljs-number">-2</span>] &gt; <span class="hljs-number">1</span>) <span class="hljs-built_in">stack</span>[++top] = i<span class="hljs-number">-2</span>;
                    &#125;
                    ans -= equal[i<span class="hljs-number">-1</span>];
                    ans += equal[i];
                    <span class="hljs-keyword">if</span>(equal[i<span class="hljs-number">-1</span>] == <span class="hljs-number">0</span>)&#123;
                        --equal[ <span class="hljs-built_in">stack</span>[top] ];
                        ans += i<span class="hljs-number">-1</span> - <span class="hljs-built_in">stack</span>[top];
                        <span class="hljs-keyword">if</span>(equal[ <span class="hljs-built_in">stack</span>[top] ] &lt;= <span class="hljs-number">1</span>) --top;
                    &#125;
                &#125;
                <span class="hljs-built_in">cout</span> &lt;&lt; ans &lt;&lt; <span class="hljs-string">&quot; &quot;</span>;
            &#125;
        &#125;
        <span class="hljs-built_in">cout</span> &lt;&lt; <span class="hljs-string">&quot;\n&quot;</span>;
    &#125;
    <span class="hljs-keyword">return</span> <span class="hljs-number">0</span>;
&#125;</code></pre></div>

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